#### Vol. 12, No. 8, 2018

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On the relative Galois module structure of rings of integers in tame extensions

### Adebisi Agboola and Leon R. McCulloh

Vol. 12 (2018), No. 8, 1823–1886
##### Abstract

Let $F$ be a number field with ring of integers ${O}_{F}$ and let $G$ be a finite group. We describe an approach to the study of the set of realisable classes in the locally free class group $Cl\left({O}_{F}G\right)$ of ${O}_{F}G$ that involves applying the work of McCulloh in the context of relative algebraic $K$ theory. For a large class of soluble groups $G$, including all groups of odd order, we show (subject to certain mild conditions) that the set of realisable classes is a subgroup of $Cl\left({O}_{F}G\right)$. This may be viewed as being a partial analogue in the setting of Galois module theory of a classical theorem of Shafarevich on the inverse Galois problem for soluble groups.

##### Keywords
Galois module structure, realisable classes, rings of integers, inverse Galois problem, relative K-group.
##### Mathematical Subject Classification 2010
Primary: 11R33
Secondary: 11R32, 11R65, 19F99